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Existence and multiplicity of solutions to the mean-field games model with mixed interactions

Xinfu Li, Xiangqing Liu, Juncheng Wei, Yuanze Wu

Abstract

In this paper, we consider the stationary version of the Mean-Field Games (MFG) models. Inspired by \cite{Albuquerque-Silva2020, Bieganowski-Mederski2021, Lin-Wei05, Mederski-Schino2021}, we develop the minimization method on the Pohozaev manifold introduced in \cite{Soave20JDE, Soave20JFA} for the existence theory of the stationary version of the Mean-Field Games (MFG) models with $2$-homogeneous hamiltonians and mixed interactions. As applications, we prove the existence and multiplicity of radial solutions of the Mean-Field Games (MFG) models with general $p$-homogeneous hamiltonians and mixed interactions under more general conditions, some of which are even new for $2$-homogeneous hamiltonians. We hope that our techniques and ideas introduced in this paper would be helpful in understanding the optimal value of the total mass in the existence theory of radial solutions to the Mean-Field Games (MFG) models with general $p$-homogeneous hamiltonians and mixed interactions, as well as that of other models.

Existence and multiplicity of solutions to the mean-field games model with mixed interactions

Abstract

In this paper, we consider the stationary version of the Mean-Field Games (MFG) models. Inspired by \cite{Albuquerque-Silva2020, Bieganowski-Mederski2021, Lin-Wei05, Mederski-Schino2021}, we develop the minimization method on the Pohozaev manifold introduced in \cite{Soave20JDE, Soave20JFA} for the existence theory of the stationary version of the Mean-Field Games (MFG) models with -homogeneous hamiltonians and mixed interactions. As applications, we prove the existence and multiplicity of radial solutions of the Mean-Field Games (MFG) models with general -homogeneous hamiltonians and mixed interactions under more general conditions, some of which are even new for -homogeneous hamiltonians. We hope that our techniques and ideas introduced in this paper would be helpful in understanding the optimal value of the total mass in the existence theory of radial solutions to the Mean-Field Games (MFG) models with general -homogeneous hamiltonians and mixed interactions, as well as that of other models.

Paper Structure

This paper contains 9 sections, 15 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. Background.
  3. Main results.
  4. Further remarks.
  5. Another background of (\ref{['e1.1']}) in the semilinear case $p=2$
  6. Organization of the paper and Notations
  7. Preliminaries
  8. The existence and multiplicity theory of (\ref{['e1.1']}) for $\mu<\mu_{a}^*$
  9. The existence of ground-state solutions

Key Result

Theorem 1.1

(Soave20JDE) Let $N\ge 3$, $2<q_1<2+\frac{4}{N}<q_2<2^*$ and $a,\mu>0$. If where $C_{N,q}:=C_{N,2,q}$ is the optimal constant of the Gagliardo-Nirenberg inequality, then the equation e1.1 in the semilinear case $p=2$ has a ground-state solution $\hat{u}_{a,\mu}$ and a mountain-pass solution $\tilde{u}_{a,\mu}$.

Theorems & Definitions (26)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Theorem 1.7
  • Remark 1.8
  • Definition 1.9
  • Theorem 1.10
  • ...and 16 more